Toothpick Squares Write- Up
The open ended problem that I have been working on is called "Toothpick Squares". It's a sort of pattern type problem, where we have to ontinue the pattern of well... toothpick squares. The first pattern is a square made of four lines that arent connected, but are in the shape of a square. In the next part of the pattern, its four of these squares, but they share the connecting toothpick lines. The next stage of the pattern is the same concept but with nine of the squares, and so on. The problem asks how many toothpicks would be in the one hundredth pattern of the problem. It also asks how many toothpicks are in the nth pattern. From the outside, the problem seemed really difficult to draw out or make a visual to, and I wasn't sure what I was going to do from a first glance. But after analyzing the problem and thinking of some ways to solve it, the problem became much more easy.
My first thought to solving this problem was to start with a visual. Which is exactly what I did. Using actual toothpicks, my and I started replicating the first 3 stages of the problem. We wrote down some ideas of how to continue the problem in our notes, and made a diagram all the way up to the sixth stage on our paper. We also tried making up some kind of formula so we could make the pattern go all the way up to one hundred without having to draw it all out. We had a few mathematicians blocks where our group had no idea what to do next, but the visuals really helped us think of the problem in a new way then just lines and numbers on paper, and it was very helpful.
We came up with an easy formula to keep the problem going in the end. After all the ways we tried to do the problem, we came up with a clean, easy formula that shows how many boxes made out of toothpicks in each stage of the pattern. The formula was the number stage of the pattern (x) squared equals the number of boxes (y). So it's x² = y. Its an easy formula that works every time. Now for the answer to how many boxes are in the hundredth stage. All we had to do was plug in 100 to x and square it to get y. The answer was ten thousand. As for how many boxes are in the xth pattern, all you have to do is plug in the number stage of the pattern into x. Easy as that.
All in all, this was a interesting problem to work on. It was similar to the pattern work we were doing before, but more in depth and it made you think more. The Habit of Mind that I thought was important for this open ended problem was significance. I think it was good that we learned how to use formulas to figure out patterns without having to draw them out, which makes doing the hundredth stage of a pattern much easier and simpler to do, and takes off a lot of the stress of making a small mistake. I think my group also learned a lot from this and I enjoyed the visuals we did, even if we didn't use it to find our final solution. I'm excited to do more open ended problems like this one or new ones in the future.
My first thought to solving this problem was to start with a visual. Which is exactly what I did. Using actual toothpicks, my and I started replicating the first 3 stages of the problem. We wrote down some ideas of how to continue the problem in our notes, and made a diagram all the way up to the sixth stage on our paper. We also tried making up some kind of formula so we could make the pattern go all the way up to one hundred without having to draw it all out. We had a few mathematicians blocks where our group had no idea what to do next, but the visuals really helped us think of the problem in a new way then just lines and numbers on paper, and it was very helpful.
We came up with an easy formula to keep the problem going in the end. After all the ways we tried to do the problem, we came up with a clean, easy formula that shows how many boxes made out of toothpicks in each stage of the pattern. The formula was the number stage of the pattern (x) squared equals the number of boxes (y). So it's x² = y. Its an easy formula that works every time. Now for the answer to how many boxes are in the hundredth stage. All we had to do was plug in 100 to x and square it to get y. The answer was ten thousand. As for how many boxes are in the xth pattern, all you have to do is plug in the number stage of the pattern into x. Easy as that.
All in all, this was a interesting problem to work on. It was similar to the pattern work we were doing before, but more in depth and it made you think more. The Habit of Mind that I thought was important for this open ended problem was significance. I think it was good that we learned how to use formulas to figure out patterns without having to draw them out, which makes doing the hundredth stage of a pattern much easier and simpler to do, and takes off a lot of the stress of making a small mistake. I think my group also learned a lot from this and I enjoyed the visuals we did, even if we didn't use it to find our final solution. I'm excited to do more open ended problems like this one or new ones in the future.